The last 4 digits of Graham’s number

In the spring we had a lot of fun talking about Graham’s number. If you haven’t seen anything about Ggraham’s number before, you might enjoy checking out that prior blog post:

An Attempt to explain Graham’s number to kids

Also definitely check out the excellent series on Graham’s number that Numberphile did with Ron Graham!

Our talk today assumes just a tiny little bit of knowledge of Graham’s number – (1) mainly that it is an outrageously tall tower of powers of three, and (2) so large, in fact, that it is nearly impossible to even imagine how large the number actually is.

We returned to Graham’s number today because my younger son just started a new chapter about last digits in his number theory book. He’ll be learning about how to find the last digit of numbers like $3^{1000}$ and some other similar numbers. It is a neat subject and a fun way to continue to build number sense.

Right at the beginning, though, he asked me why we were only taking about the last digit – why not the tens digit, or hundreds digit? Well . . . today we’ll talk about the last 4 digits of Graham’s number just for fun.

Our first talk is a quick review of Graham’s number. If you want to understand the “up arrow” notation check out the links above, but that notation isn’t important for today. All you really need to know is that Graham’s numbers is a huge tower of powers of 3:

With the review out of the way we turn our attention to the last digit of Graham’s number. After looking at the first few powers of 3 we see that the last digit appears to repeat every 4th number. Quite surprisingly that pattern gives us enough information to infer that the last digit of Graham’s number is either 3 or 7. We spend probably half of the movie arriving at that fact and then we perform a more detailed calculation to see what the last digit actually is. One point that caused a little bit of confusion is that we need to look at the power itself in cycles of 4 (or what the remainder is when you divide by 4) even though we are looking for the last digit (so remainder when divided by 10):

Next we moved to the computer to get a little help from Mathematica! We essentially repeat the calculations that we just did on the whiteboard, but looking at the last two digits rather than looking only at the last digit. When you look at the last two digits you see a pattern that repeats every 20 powers – hence why a computer is helpful! Once we know that there is a pattern that repeats every 20 numbers we can use the computer to perform the same computation that we did by hand in the last movie to find the last two digits of Graham’s number:

The next step was looking for the last 3 digits. It is essentially the same process. We found that the last digit of the powers of 3 repeat every 4 powers and the last two digits of powers of 3 repeat every 20, so I asked the boys what they thought the pattern in the last 3 digits would be. They both guessed the repetition would be every 100 powers, which turns out to be right. Again, the computer is your friend here!

Also, we made a little mistake in this video and got confused between 100 and 1,000 when the pattern was repeating. Luckily that just made more work for the computer to do rather than for us (which is probably why we didn’t notice), but the result is unchanged (luckily).

We wrapped up by wondering why we are seeing powers of 5 in the way the digits repeat. The units digit repeats every 4 powers, the last two digits repeat every 20, the last three digits repeat every 100 powers, and the last 4 repeat every 500 powers – why are we multiplying by 5 every step? We didn’t arrive at an answer for this problem, but rather left it as something to wonder about.

The last thing we did was check out the Wikipedia page about Graham’s number to see if we got the last three digits right. That page gives the last 500 digits and our last 3 do actually match! We also now have a procedure to use to (perhaps) find all 500 digits.

So, a fun little project. Kicking myself for the 100 vs 1,000 mistake, but I guess that happens. The project kept the kids engaged all the way through – both the math and the computer results are really interesting. It is amazing (especially for kids) to see that even though you can’t really say anything at all about the number itself, you can compute some of the final digits.